ON VALIDITY OF THE ASYMPTOTIC EXPANSION APPROACH IN...

The Annals of Applied Probability2003, Vol. 13, No. 3, 914–952© Institute of Mathematical Statistics, 2003

ON VALIDITY OF THE ASYMPTOTIC EXPANSION APPROACHIN CONTINGENT CLAIM ANALYSIS

BY NAOTO KUNITOMO AND AKIHIKO TAKAHASHI

University of Tokyo

Kunitomo and Takahashi (1995, 2001) have proposed a new methodology,called small disturbance asymptotics, for the valuation problem of financialcontingent claims when the underlying asset prices follow a general classof continuous Itô processes. It can be applicable to a wide range ofvaluation problems, including complicated contingent claims associated withthe Black–Scholes model and the term structure model of interest rates in theHeath–Jarrow–Morton framework. Our approach can be rigorously justifiedby an infinite-dimensional analysis called the Watanabe–Yoshida theory onthe Malliavin calculus recently developed in stochastic analysis.

1. Introduction. In the past decades various contingent claims includingfutures, options, swaps and other derivative securities have been introduced andactively traded in financial markets. Except some simple cases such as theoriginal Black–Scholes model in which the underlying assets follow the geometricBrownian motions and the risk free rate is constant, however, it has been difficultto derive the explicit formulas for the fair market values of financial contingentclaims. Meanwhile, Kunitomo and Takahashi (1995, 2001), and Takahashi (1999)have presented a new methodology called the small disturbance asymptotic theorywhich is widely applicable to the valuation problem of financial contingent claimswhen the underlying asset prices follow the general class of continuous Itôprocesses. They have given rather simple formulas which are useful for variousvaluation problems of contingent claims in financial economics.

For the Black–Scholes economy, Takahashi (1999) has systematically investi-gated the valuation problem of various contingent claims when the vector of d assetprices St = (Si

t ) (i = 1, . . . , d; 0 ≤ t ≤ T < +∞) follows the stochastic differen-tial equation

Sit = Si

0 +∫ t

0µi∗(Sv, v) dv +

m∑j=1

∫ t

0σ ij∗ (Sv, v) dwj

v,(1.1)

where d × 1 vector µ∗(Sv, v) = (µi∗(Sv, v)) and d × m matrix σ∗(Sv, v) =(σ

ij∗ (Sv, v)) are the instantaneous mean and the volatility functions, respectively,and {wi

v} are Brownian motions. In this Black–Scholes economy, we have to

Received May 2000; revised February 2002.AMS 2000 subject classifications. Primary 90A09; secondary 60H07.Key words and phrases. Valuation of financial contingent claims, asymptotic expansion, small

disturbance asymptotics, validity, Watanabe–Yoshida theory, Malliavin calculus.

914

CONTINGENT CLAIM ANALYSIS 915

change the underlying measure because of the no-arbitrage theory in finance [see,e.g., Chapter 6 of Duffie (1996) on the standard theory]. Then we can consider thesituation when S

(ε)t satisfies

S(ε)t = S0 +

∫ t

0r(S(ε)

v , v)S(ε)

v dv + ε

∫ t

0σ(S(ε)

v , v)dwv,(1.2)

where S(ε)t [= (S

(ε)it )] is a d × 1 vector with the parameter ε (0 < ε ≤ 1),

σ(S(ε)v , v) (d ×m) is the volatility term, r(·, ·) is the risk free (positive) interest rate,

and wv [= (wiv)] is an m×1 vector. The small disturbance asymptotic theory under

the no-arbitrage theory can be constructed by considering the situation when ε → 0and we can develop the valuation method of contingent claims based on {S(ε)

t }.

NOTE. The limit of stochastic process S(ε)t is the solution of an ordinary

differential equation when ε → 0 in this formulation. There can be an alternativeformulation such that the limit is the solution of a stochastic differential equation.See Kim and Kunitomo (1999), Kunitomo and Kim (2001), Sørensen andYoshida (2000) or Takahashi and Yoshida (2001) on this formulation and someapplications in financial problems. However, it requires a set of different argumentsincluding the partial Malliavin covariances.

For the term structure model of interest rates in the HJM framework [Heath,Jarrow, and Morton (1992)], let P (s, t) denote the price of the discount bond at s

with maturity date t (0 ≤ s ≤ t ≤ T < +∞). When it is continuously differentiablewith respect to t and P (s, t) > 0 for 0 ≤ s ≤ t ≤ T, the instantaneous forwardrate at s for future date t (0 ≤ s ≤ t ≤ T ) is given by f (s, t) = − ∂ log P(s,t)

∂t. The

no-arbitrage condition requires the drift restrictions on a family of forward ratesprocesses {f (s, t)} for 0 ≤ s ≤ t ≤ T to follow the stochastic integral equation

f (s, t) = f (0, t)

+∫ s

0

m∑i=1

[σ i∗(f (v, t), v, t

) ∫ t

vσ i∗(f (v, y), v, y

)dy

]dv

+m∑

i=1

∫ s

0σ i∗(f (v, t), v, t

)dwi

v,

(1.3)

where f (0, t) are nonrandom initial forward rates, {wiv; i = 1, . . . ,m} are

m Brownian motions and {σ i∗(f (v, t), v, t); i = 1, . . . ,m} are the volatilityfunctions. When f (s, t) is continuous at s = t for 0 ≤ s ≤ t ≤ T, the instantaneousspot interest rate process can be defined by r(t) = lims→t f (s, t). In thisframework of stochastic interest rate economy, Kunitomo and Takahashi (1995,2001) have investigated the valuation of contingent claims when a family of

916 N. KUNITOMO AND A. TAKAHASHI

forward rate processes obey

f (ε)(s, t) = f (0, t)

+ ε2∫ s

0

m∑i=1

[σ i(f (ε)(v, t), v, t

) ∫ t

vσ i(f (ε)(v, y), v, y

)dy

]dv

+ ε

m∑i=1

∫ s

0σ i(f (ε)(v, t), v, t

)dwi

v,

(1.4)

where 0 < ε ≤ 1. The volatility functions {σ i(f (ε)(s, t), s, t); i = 1, . . . ,m} de-pend not only on s and t , but also on f (ε)(s, t) in the general case. The instanta-neous spot interest rate process can be defined by r(ε)(t) = lims→t f

(ε)(s, t). Thenthe small disturbance asymptotic theory can be constructed by considering the sit-uation when ε → 0 and we can develop the valuation method of contingent claimsbased on {f (ε)(s, t)} and the discount bond prices

P (ε)(t, T ) = exp[−∫ T

tf (ε)(t, u) du

].(1.5)

The main purpose of this paper is to give the validity of the asymptoticexpansion approach along the line called the Watanabe–Yoshida theory on theMalliavin calculus recently developed in stochastic analysis. The Malliavincalculus has been developed as an infinite-dimensional analysis of Wienerfunctionals by several probablists in the last two decades. We intend to apply thispowerful calculus on continuous stochastic processes to the valuation problemof financial contingent claims along the line developed by Watanabe (1987)and subsequently by Yoshida (1992). However, the continuous-time stochasticprocesses appearing in financial economics are not necessarily time-homogeneousMarkovian in the usual sense while the existing asymptotic expansion methodsinitiated by Watanabe (1987) and refined by Yoshida (1992) have been developedfor time-homogeneous Markovian processes. Hence we need to extend some ofthe existing results on the validity of the asymptotic expansion approach. Also themathematical devices used in the Watanabe–Yoshida theory have not been standardfor finance as well as in many applied fields due to the recent mathematicaldevelopments involved. In this paper we intend to give a rigorous discussion onthe validity of the asymptotic expansion approach in a unified way. Althoughsome of the following derivations have been already reported in Kunitomo andTakahashi (1995, 2001) and Takahashi (1999), these papers did not give manyimportant proofs on the validity of the asymptotic expansion approach.

In this paper we shall also illustrate the usefulness of the asymptotic expansionapproach by showing some numerical examples. Since several related papers havealready appeared [Kunitomo and Takahashi (2001) and Takahashi (1999), e.g.],we shall only discuss simple examples with analytical difficulties from otherapproaches.


In Section 2, we give some preliminary mathematical devices, that will beneeded in the following derivations. Section 3 is on the validity of our approachfor the continuous Markovian setting, while Section 4 is on the validity of ourapproach for the HJM setting of the interest rates model. We give some numericalexamples in Section 5 and concluding remarks in Section 6. Some mathematicaldetails will be given in the Appendix.

2. Preliminary mathematics. We shall first prepare the fundamental results,including Theorem 2.2 of Yoshida (1992), which is in turn a truncated version ofTheorem 2.3 of Watanabe (1987). The theory by Watanabe (1987) on the Malliavincalculus and Theorem 2.2 of Yoshida (1992) are the fundamental ingredients toshow the validity of our asymptotic expansion method. This is the reason why wecall it the Watanabe–Yoshida theory on the Malliavin calculus. For our purpose, weshall freely use the notation by Ikeda and Watanabe (1989) as a standard textbook.The interested reader should see Watanabe (1984, 1987), Ikeda and Watanabe(1989), Yoshida (1992, 1997), Shigekawa (1998) or Nualart (1995).

2.1. Some notation and definitions. Let W be the m-dimensional Wienerspace, which is a Banach space consisting of the totality of continuous functionsw : [0, T ] → Rm [w(0) = 0] with the topology induced by the norm ‖w‖ =max0≤t≤T |w(t)|. Let also H be the Cameron–Martin subspace of W , whereh(t) = (hj (t)) ∈ H is in W and is absolutely continuous on [0, T ] with squareintegrable derivative h(t) endowed with the inner product defined by

〈h1, h2〉H =m∑

j=1

∫ T

0h

j1(s)h

j2(s) ds.(2.1)

We shall use the notation of the H -norm as |h|2H = 〈h,h〉H for any h ∈ H .A function f : W → R is called a polynomial functional if there exist n ∈ N ,h1, h2, . . . , hn ∈ H and a real polynomial p(x1, x2, . . . , xn) of n-variables suchthat f (w) = p([h1](w), [h2](w), . . . , [hn](w)) for hi = (h

ji ) ∈ H , where

[hi](w) =m∑

j=1

∫ T

0h

ji dwj .(2.2)

are defined in the sense of Itô’s stochastic integrals.The standard Lp-norm of the R-valued Wiener functional F is defined by

‖F‖p = (∫W |F |pP (dω))1/p. Also a sequence of the norms of the R-valued

Wiener functional F for any s ∈ R, and p ∈ (1,∞) is defined by

‖F‖p,s = ‖(I − L)s/2F‖p,(2.3)

where L is the Ornstein–Uhlenbeck operator and ‖·‖p is the Lp-norm in thestochastic analysis. The O–U operator in (2.3) means that (I − L)s/2F =∑∞

n=0(1 + n)s/2JnF , where Jn are the projection operators in the Wiener


homogeneous chaos decomposition in L2(R). They are constructed by the totalityof R-valued polynomials of degree at most n, denoted by P n.

Let P (R) denote the totality of R-valued polynomials on the Wiener space(W ,P ). Then P (R) is dense in Lp(R) and can be extended to the totality ofsmooth functionals S (the C∞ functions with derivatives of polynomial growthorders). Then we can construct the Banach space Ds

p(R) as the completion ofP (R) with respect to ‖ · ‖p,s . The dual space of Ds

p(R) is D−sq (R), where s ∈ R,

p > 1 and 1/p + 1/q = 1. The space D∞(R) = ⋂s>0

⋂1<p<+∞ Ds

p(R) is the

set of Wiener functionals and D−∞

(R) =⋃s>0

⋂1<p<+∞ D−s

p (R) is a space ofgeneralized Wiener functionals. For F ∈ P (R) and h ∈ H , the derivative of F inthe direction of h is defined by

DhF(w) = limε→0

1

ε{F(w + εh) − F(w)}.(2.4)

Then for F ∈ P (R) and h ∈ H there exists DF ∈ P (H ⊗R) such that DhF(w) =〈DF(w),h〉H, where 〈·〉H is the inner product of H and DF is called theH -derivative of F. Also for F ∈ S(R) there exists a unique DF ∈ S(H ⊗ R).

More generally, for a separable Hilbert space E, a function f : W → E iscalled a polynomial functional if there exist n ∈ N , h1, h2, . . . , hn ∈ H and realpolynomials pi(x1, x2, . . . , xn) of n-variables such that

f (w) =d∑

i=1

pi

([h1](w), [h2](w), . . . , [hn](w))ei

for some d ∈ N , where e1, . . . , ed ∈ E. The totality of E-valued polynomialfunctions and the totality of E-valued smooth functionals are denoted by P (E)

and S(E), respectively. By extending the above construction for P (R) to S(E),there exists DF ∈ S(H ⊗ E) such that DhF(w) = 〈DF(w),h〉H, where 〈·〉H isthe inner product of H .

By repeating this procedure, we can sequentially define the kth order H -deriva-tive DkF ∈ S(H⊗k ⊗ E) for k ≥ 1, and it is known that the norm ‖ · ‖p,s isequivalent to the norm

∑sk=0 ‖Dk · ‖p . In particular, for F = (F i) ∈ D1

p(Rd), wedefine the Malliavin covariance by

σMC(F ) = (〈DF i(w),DF j(w)〉H ), i, j = 1, . . . , d.(2.5)

2.2. Asymptotic expansions. Let X(ε)(w) = (Xi(ε)(w)) (i = 1, . . . , d ; ε ∈(0,1]) be a Wiener functional with a parameter ε. Then we need to define theasymptotic expansion of X(ε)(w) with respect to ε in the proper mathematicalsense. For k > 0, X(ε)(w) = O(εk) in Ds

p(E) as ε ↓ 0 means that

lim supε↓0

‖X(ε)‖p,s

εk< +∞.(2.6)


If for all p > 1, s > 0 and every k = 1,2, . . . ,

X(ε)(w) − (g1 + εg2 + · · · + εk−1gk) = O(εk)(2.7)

in Dsp(E) as ε ↓ 0, then we say that X(ε)(w) has an asymptotic expansion

X(ε)(w) ∼ g1 + εg2 + · · ·(2.8)

in D∞(E) as ε ↓ 0 with g1, g2, . . . ∈ D∞(E).Also if for every k = 1,2, . . . , there exists s > 0 such that, for all p > 1,

X(ε)(w), g1, g2, . . . ∈ D−sp (E) and

X(ε)(w) − (g1 + εg2 + · · · + εk−1gk) = O(εk)(2.9)

in D−sp (E) as ε ↓ 0, then we say that X(ε)(w) ∈ D

−∞(E) has an asymptotic

expansion

X(ε)(w) ∼ g1 + εg2 + · · ·(2.10)

in D−∞

(E) as ε ↓ 0 with g1, g2, . . . ∈ D−∞

(E).Let S(Rd) be the totality of C∞ rapidly decreasing functions on Rd and S ′(Rd)

be its dual. Also let ηε ∈ D∞(R) and ψ(y) be a smooth function such that0 ≤ ψ(y) ≤ 1 for y ∈ R, ψ(y) = 1 for |y| ≤ 1/2 and ψ = 0 for |y| ≥ 1. It isknown that if for any p > 1 the Malliavin covariance of X(ε) ∈ D∞(Rd) satisfies

supε∈(0,1]

E[1{|ηε |≤1}

(det[σMC(X(ε))])−p

]< ∞,(2.11)

the composite functional G = ψ(ηε)G ◦ X(ε) ∈ D−∞

(Rd) is well defined withany G ∈ S ′(Rd). Then the coupling

D−∞

⟨G, J

⟩D∞ =

D−∞

⟨G(X(ε)),ψ(ηε)J

⟩D∞,(2.12)

for any J ∈ D∞(Rd), is well defined and we can use the notation of theexpectation E[ψ(ηε)G(X(ε))] by taking J = 1. With these formulations andnotation we are ready to state a simplified version of Theorem 2.2 of Yoshida(1992), which is a truncated version of Theorem 2.3 of Watanabe (1987). Thevalidity of the asymptotic expansion is obtained by showing that the conditions ofthe next theorem are met in our situations.

THEOREM 2.1 [Yoshida (1992)]. Suppose the following set of sufficientconditions are satisfied:

(i) {X(ε)(w); ε ∈ (0,1]} ∈ D∞(Rd) with X(ε)(w) = (Xi(ε)(w)).(ii) X(ε)(w) has the asymptotic expansion X(ε)(w) ∼ g1 + εg2 + · · · in

D∞(Rd) as ε ↓ 0 with g1, g2, . . . ∈ D∞(Rd).(iii) {ηε(w); ε ∈ (0,1]} ⊂ D∞(R) and it is O(1) in D∞(R) as ε ↓ 0.


(iv) For any p > 1,

supε∈(0,1]

E[1{|ηε|≤1}

(det[σMC(X(ε))])−p

]< ∞.(2.13)

(v) For any k ≥ 1,

limε→0

ε−kP{|ηε| > 1

2

}= 0.(2.14)

(vi) Let φ(ε)(x) be a smooth function in (x, ε) on Rd × (0,1] with allderivatives of polynomial growth order in x uniformly in ε.

Then ψ(ηε)φ(ε)(X(ε))IB(X(ε)) has an asymptotic expansion,

ψ(ηε)φ(ε)(X(ε))IB(X(ε)) ∼ �0 + ε�1 + · · ·(2.15)

in D−∞

(R) as ε ↓ 0, where IB is the indicator function for any Borel set B and�0,�1, . . . are determined by the formal Taylor expansion with respect to X(ε)

in (2.10).

REMARK. We have to mention an intuitive meaning of the asymptoticexpansion in the above theorem. If we truncate the random variable under thecondition of (2.13), then the asymptotic expansion in (2.15) implies

lim supε↓0

∣∣∣∣ 1

εkE[ψ(ηε

c)φ(ε)(X(ε))IB(X(ε))

− (�0 + ε�1 + · · · + εk−1�k−1)]∣∣∣∣< +∞

for any integer k ≥ 1 if we use the expectation operation in the propermathematical sense. The calculations of the generalized expectation operationsfor the generalized Wiener functionals will be discussed in Section 3.

3. Validity in the Black–Scholes economy. Let (,F ,Q, {Ft }t∈[0,T ]) be thefiltered probability space with T < +∞. For ε ∈ (0,1] and 0 < t ≤ T , the vectorof d security prices follow a sequence of stochastic differential equations

S(ε)t = S0 +

∫ t

0µ(S(ε)

s , s)ds +

∫ t

0εσ(S(ε)

s , s)dws,(3.1)

where µ(S(ε)s , s) = r(S

(ε)s , s)S

(ε)s and σ(S

(ε)s , s) = (σ ij (S

(ε)s , s)) are Rd ×

[0, T ] → Rd and Rd ×[0, T ] → Rd ⊗Rm Borel measurable functions in (S(ε)s , s),

respectively, and ws [= (wis)] are the vector of m × 1 Brownian motions with re-

spect to {Ft}. We further assume that the drift and the volatility functions are con-tinuous and C∞ for s ∈ [0, T ] with bounded derivatives of any order in the first


argument. That is, for the first argument there exist positive constants M1(k) andM2(k) (k ≥ 1) such that for any i = 1, . . . , d and j = 1, . . . ,m,

supS∈Rd,0≤s≤T

∣∣∣∣∣ ∂kµi(S(ε)s , s)

∂Si1(ε)s · · · ∂S

ik(ε)s

∣∣∣∣∣< M1(k),(3.2)

supS∈Rd,0≤s≤T

∣∣∣∣ ∂kσ ij (S(ε)s , s)

∂Si1(ε)s · · · ∂S

ik(ε)s

∣∣∣∣< M2(k),(3.3)

where µ(S(ε)s , s) = (µi(S

(ε)s , s)), and we shall denote the partial derivatives as

∂ki1,...,ik

µi(S(ε)s , s) and ∂k

i1,...,ikσ ij (S

(ε)s , s), respectively. We further consider the

case that there exists a positve M3 such that

sup0≤s≤T

[|µ(0, s)| + |σ(0, s)|]< M3,(3.4)

where the notation |A| =√∑

i,j |aij |2 for any matrix A = (aij ) and |a| =√∑i |ai |2 for any vector a = (ai) are used. These conditions imply that there

exists some positive Ki > 0 (i = 1,2) such that for all s, t ∈ [0, T ],∣∣µ(S(ε)s , s

)∣∣+ ∣∣σ (S(ε)s , s

)∣∣< K1(1 + |S(ε)

s |),(3.5) ∣∣µ(S(ε)1s , s

)− µ(S

(ε)2s , s

)∣∣+ ∣∣σ (S(ε)1s , s

)− σ(S

(ε)2s , s

)∣∣< K2∣∣S(ε)

1s − S(ε)2s

∣∣.(3.6)

Then the standard argument in stochastic analysis shows the existence of theunique strong solution which has continuous sample paths and is in Lp(Rd) forany 1 < p < ∞. In the remainder of this section, we will mainly discuss thevalidity of the asymptotic expansion of φ(X

(ε)T )IB(X

(ε)T ) for any Borel set B,

where X(ε)T is defined by

X(ε)T = 1

ε

(S

(ε)T − S

(0)T

)(3.7)

and S(0)T is the solution of the ordinary differential equation

S(0)T = S0 +

∫ T

0µ(S(0)

s , s)ds.(3.8)

For illustrations in this section, we only mention simple examples. When we taked = m = 1, φ(x) = (x +y) and IB(x) = {x ≥ −y} for a constant y, it correspondsto the valuation problem of European options in mathematical finance. We shallgive another example for the Asian options, which was considered by Kunitomoand Takahashi (1992).

First, we shall show that S(ε)T is a smooth Wiener functional in the sense

of Malliavin. A more detailed proof when d = m = 1 has been discussed byKunitomo and Takahashi (1998) and Takahashi (1999).


THEOREM 3.1. Under the assumptions in (3.1)–(3.3), S(ε)T is in D∞(Rd) and

has an asymptotic expansion

S(ε)T ∼ S

(0)T + εg1T + ε2g2T + · · ·(3.9)

as ε ↓ 0 with g1T , g2T , . . . ∈ D∞(Rd).

PROOF. (i) The first part of our proof is to show that S(ε)T is in D∞(Rd).

But it is well known that S(ε)T ∈ D∞(Rd) when S

(ε)T follows a time-homogeneous

Markovian process. Since any time-dependent Markovian process can be repre-sented as a time-homogeneous Markovian process, we can immediately applythe general result to our case. [See Chapter V of Ikeda and Watanabe (1989), orKusuoka and Strook (1982).]

(ii) We shall prove the second part of Theorem 3.1. The coefficientsappearing in the asymptotic expansion of S

(ε)T are given by the formal Taylor

formula. By expanding X(ε)T as X

(ε)T = g1T + εg2T + ε2g3T +· · · with respect to ε,

we can determine the coefficients {gjT (j ≥ 1)} recursively. The ith component ofthe leading term (i = 1,2, . . . , d) is given by

g(i)1T =

d∑l1=1

∫ T

0∂l1µ

i(S(0)

s , s)g

(l1)1s ds +

m∑j ′=1

∫ T

0σ ij ′(

S(0)s , s

)dwj ′

s .

Then it can be written as

g(i)1T =

d∑j=1

m∑j ′=1

∫ T

0(YT Y−1

s )ij σ jj ′(S(0)

s , s)dwj ′

s ,

where Yt = Y(0)t is the solution of the ordinary differential equation dY il =∑d

k=1 ∂kµi(S

(0)t , t)Y kl dt . This equation can be solved and its solution is written

as Yt = exp(∫ t

0 (∂jµi(S

(0)s , s)) ds) with the initial condition Y0 = Id .

For n ≥ 2, we recursively define the ith component of each term g(i)nT by

g(i)nT =

n∑k=1

m1+···+mk=n

∫ T

0

[1

k!d∑

l1,...,lk=1

∂kl1,...,lk

µi(S(0)

s , s) k∏j=1

g(lj )mj ,s

]ds

+n∑

k=1m1+···+mk=n−1

∫ T

0

[1

k!d∑

l1,...,lk=1

m∑j ′=1

∂kl1,...,lk

σ ij ′(S(0)

s , s) k∏j=1

g(lj )mj ,s dwj ′

s

],

where mj (j = 1, . . . , k) are positive integers.

By the boundedness of YT ,Y−1s , σ (S

(0)s , s) on [0, T ], we have E[|g1s |p] < ∞,

s ∈ [0, T ] for any 1 < p < ∞. Given g1s ∈ Lp(Rd), we have E[|g2s |p] < ∞ for


any 1 < p < ∞. By the same token, the relation gks ∈ Lp(Rd) can be obtainablerecursively given gjs ∈ Lp(Rd) (j = 1,2, . . . k − 1) and we have g1T , g2T , . . . ∈⋂

1<p<∞ D1p(Rd).

Next, we note that

Dhg(i)1T =

d∑j=1

m∑j ′=1

∫ T

0(YT Y−1

s )ij σ jj ′(S(0)

s , s)hj ′

s ds.

For higher order derivatives we use an induction argument and we assumegnT ∈ ⋂

1<p<∞ Dkp(Rd) (k ≥ 1) for any n ≥ 1. Then we need to show that

gnT ∈ ⋂1<p<∞ Dk+1

p (Rd). Actually, we can show the Lp-boundedness of anyorder H -derivatives of gnT (n ≥ 1) recursively. In our evaluations of higher orderH -derivatives, we need a version of Burkholder’s inequality for Hilbert spacevalued stochastic integrals proved by Lemma 2.1 of Kusuoka and Strook (1982).Given g1s ∈ D∞(Rd) for any s ∈ [0, T ] , we can recursively show that gnT ∈D∞(Rd).

(iii) Finally, for any n (n ≥ 1) and s ∈ [0, T ] let

Z(ε)ns = 1

εn

[X(ε)

s − g1s − εg2s − · · · − εn−1gns

].

By using (3.1) repeatedly and applying the standard arguments, we can show thatZ

(ε)1s ∈ Lp(Rd) for any p > 1 uniformly with respect to ε. Again, by applying

recursive arguments and using induction with respect to n, we can show thatZ

(ε)ns and their H -derivatives are in Lp uniformly with respect to ε after tedious

arguments, which were omitted. �

We now return to the original problem on the normalized random variable X(ε)T

in (3.7). Using Theorem 3.1, we see X(ε)T is in D∞(Rd) and has a proper

asymptotic expansion

X(ε)T ∼ g1T + εg2T + · · ·

in D∞(Rd) with g1T , g2T , . . . ∈ D∞(Rd). By using the Fubini-type result in oursetting for any h ∈ H , the first-order H -derivative DhS

(ε)T satisfies

DhS(ε)iT =

m∑j=1

d∑k=1

∫ T

0ε∂kσ

ij(S(ε)

s , s)DhS

(ε)ks dwj(s)

+d∑

k=1

∫ T

0∂kµ

i(S(ε)s , s

)DhS

(ε)ks ds +

m∑j=1

∫ T

0εσ ij (S(ε)

s , s)hj

s ds.

Then it can be represented as

DhS(ε)iT =

d∑j=1

m∑j ′=1

∫ T

0

(Y

(ε)T Y (ε)−1

s

)ijεσ jj ′(

S(ε)s , s

)hj ′

s ds,(3.10)


where Y(ε)t is the solution of the stochastic differential equation

dY(ε)ilt =

d∑k=1

∂kµi(S

(ε)t , t

)Y

(ε)klt dt + ε

m∑j=1

d∑k=1

∂kσij(S

(ε)t , t

)Y

(ε)klt dw

jt(3.11)

and Y(ε)0 = Id . Then the Malliavin covariance of the normalized random variable

σMC(X(ε)T ) = (σ

ijMC(X

(ε)T )) is given by

σijMC

(X

(ε)T

)= m∑k=1

∫ T

0

[Y

(ε)T Y (ε)−1

s σ(S(ε)

s , s)]ik[

Y(ε)T Y (ε)−1

s σ(S(ε)

s , s)]jk

ds.

We shall consider the uniform nondegeneracy of the Malliavin covariance, whichis the most important step in the application of Theorem 2.1. For this purpose, weneed the following assumption.

ASSUMPTION I. For any T > 0 the d × d matrix g1 = (ijg1) is positive

definite, where ijg1 is given by

ijg1

=m∑

k=1

∫ T

0

[YT Y−1

s σ(S(0)

s , s)]ik[

YT Y−1s σ

(S(0)

s , s)]jk

ds.(3.12)

This assumption assures the nondegeneracy of the limiting distribution of therandom variable, which can be easily checked in applications. We define ηε

c as

ηεc = c

∫ T

0

∣∣Y (ε)T

(Y (ε)

s

)−1σ(S(ε)

s , s)− YT Y−1

s σ(S(0)

s , s)∣∣2 ds(3.13)

for any c > 0. Let ξ(ε)s,t = Y

(ε)t (Y ε

s )−1σ(S(ε)s , s) and ξs,t = YtY

−1s σ (S

(0)s , s) and

we note an inequality |ξ (ε)s,T ξ

(ε)∗s,T − ξs,T ξ∗

s,T | ≤ |ξ (ε)s,T − ξs,T |2 + 2|ξs,T ||ξ (ε)

s,T − ξs,T |,where the notation A∗ is used for the transpose of any matrix A = (aij ). Then

condition |ηδc | ≤ 1 is equivalent to

∫ T0 |ξ (ε)

s,T − ξs,T |2 ds ≤ 1/c and we have

∣∣σMC(X

(ε)T

)− g1

∣∣≤ 1

c+ 2

∣∣g1

∣∣√

1

c

for |ηεc | ≤ 1. Thus we can take c0 such that for any c > c0 > 0, 0 < g1 +

(σMC(X(ε)T ) − g1) = σMC(X

(ε)T ) holds uniformly for ε ∈ (0,1]. Hence we have

the next result on the uniform nondegeneracy of the Malliavin covariance.

THEOREM 3.2. Under the assumptions in (3.1)–(3.3) and Assumption I, theMalliavin covariance σMC(X

(ε)T ) is uniformly nondegenerate. That is, there exists

c0 > 0 such that for c > c0 and any p > 1,

supε∈(0,1]

E[1{ηε

c≤1}{det(σMC(X

(ε)T )

)}−p]< ∞.(3.14)


By using the results in Theorem 3.1, Theorem 3.2 and Lemma A.2 in theAppendix, we have shown that the conditions of Theorem 2.1 are satisfied. Thenwe immediately obtain the next result.

THEOREM 3.3. Under the assumptions in (3.1)–(3.3) and Assumption I,for a smooth function φ(ε)(x) with all derivatives of polynomial growth orders,ψ(ηε

c)φ(ε)(X

(ε)T )IB(X

(ε)T ) has an asymptotic expansion

ψ(ηεc)φ

(ε)(X(ε)T )IB(X

(ε)T ) ∼ �0 + ε�1 + ε2�2 + · · ·(3.15)

in D−∞

(R) as ε ↓ 0, where B is a Borel set, ψ(x) is a smooth function suchthat 0 ≤ ψ(x) ≤ 1 for x ∈ R, ψ(x) = 1 for |x| ≤ 1/2, ψ = 0 for |x| ≥ 1,and �0,�1, . . . are determined by the formal Taylor expansion with respect toX

(ε)T (∼ g1T + εg2T + · · ·).Then we obtain an asymptotic expansion of the expectation of (3.15), which is

the direct consequence of Theorem 2.1 and Theorem 3.3.

COROLLARY 3.1. Under the assumptions in (3.1)–(3.3) and Assumption I, anasymptotic expansion of E[φ(ε)(X

(ε)T )IB(X

(ε)T )] is given by

E[φ(ε)

(X

(ε)T

)IB(X

(ε)T

)]∼ E[ψ(ηε

c)φ(ε)(X

(ε)T

)IB(X

(ε)T

)]∼ E[�0] + εE[�1] + ε2E[�2] + · · ·

(3.16)

as ε ↓ 0, where φ(ε)(·), ψ(·), �j (j ≥ 0) and B are defined as in Theorem 3.3.

Our next objective is to show that the resulting formulas of asymptoticexpansions are equivalent to those from the method based on the simple inversiontechnique for the characteristic function, which have been used by Kunitomo andTakahashi (1995, 2001), and Takahashi (1999). It is possible to explicitly derive theformulas of the asymptotic distribution function and the density function, and alsothose of the expectations of the random variables involving X

(ε)T in certain ranges.

We start with the explicit evaluation of each terms appearing in Theorem 3.3. Weobserve that the first term in Theorem 3.3 is given by

�0 = φ(0)(g1T )IB(g1T ).(3.17)

Then by applying the Taylor expansion, for any n ≥ 1, we have

�n =

m1+···+mk=m+2k

m′1+···+m′

k′=m′+2k′∑k,l,m,k′,m′≥0

k+l+m+k′+m′=n

d∑l′1,...,l

′k′=1

1

k′!∂k′l′1,...,l

′k′IB(g1T )

k′∏j=1

g(l

′j )

m′j T

×(

d∑l1,...,lk=1

1

k!l!∂kl1,...,lk

∂lεφ

(0)(g1T )

k∏j=1

g(lj )

mj T

),

(3.18)


where mj ≥ 2 and m′j ≥ 2.

When d = m = 1 in particular, we have relatively simple and useful forms. Forinstance, the first two terms are given by

�1 =[∂φ(0)

∂ε(g1T ) + ∂φ(0)(g1T )g2

]IB(g1T ) + φ(0)(g1T )∂IB(g1T )g2T ,

�2 =[∂φ(0)

∂ε(g1T ) + ∂φ(0)(g1T )g2T

]∂IB(g1T )g2T

+[

1

2

∂2φ(0)

∂ε2(g1T ) +

{∂2φ(0)(x)

∂x ∂ε

∣∣∣∣x=g1T

}g2T

+ ∂φ(0)(g1T )g3T + 1

2∂2φ(0)(g1T )g2

2T

]IB(g1T )

+ φ(0)(g1T )

{1

2∂2IB(g1T )g2

2T + ∂IB(g1T )g3T

}.

In the above expressions the differentiations of the indicator function IB(·) haveproper mathematical meanings as the generalized Wiener functionals. As wehave indicated at the end of Section 2, the rigorous mathematical foundation ofdifferentiation and the integration by parts formula have been given in Chapter Vof Ikeda and Watanabe (1989) and Yoshida (1992, 1997). The next result summa-rizes the explicit expressions for the asymptotic expansion of expectations of theabove random variables based on the Gaussian density function up to third-orderterms.

THEOREM 3.4. In the asymptotic expansion of (3.16) terms E[�n] (n =0,1,2) are given by

E[�0] =∫B

φ(0)(x)n[x | 0,g1

]dx,

E[�1] =∫B

{∂εφ

(0)(x)n[x | 0,g1

]

− φ(0)(x)

d∑i=1

∂i

{E[g

(i)2 | g1 = x

]n[x | 0,g1

]}}dx,

E[�2] =∫B

(−∂εφ

(0)(x)

d∑i=1

∂i

{E[g

(i)2 |g1 = x

]n[x | 0,g1

]}

+ 12∂2

ε φ(0)(x)n[x | 0,g1

]


− 12

d∑i,j=1

φ(0)(x)∂2ij

{E[g

(i)2 g

(j)2 |g1 = x

]n[x | 0,g1

]}

−d∑

i=1

φ(0)(x)∂i

{E[g

(i)3 |g1 = x

]n[x | 0,g1

]})dx,

where we denote g(i)n = g

(i)nT (i = 1, . . . , d;n ≥ 1), ∂εφ

(0) = ∂φ(ε)

∂ε|ε=0(x),

E[z|g1 = x] as the conditional expectation of z given g1 = x, and n[x | 0,]as the density function of the d-dimensional Gaussian distribution with zero meanand the variance-covariance matrix .

PROOF. Without loss of generality, we only give the proof for the case whend = m = 1. The essential part of the present proof is in the fact that we can use theintegration by parts operation repeatedly. First, the formula for E [�0] is the directresult of calculation. Second, the expectation of the first term of �1 is given by

E[{

∂φ(ε)

∂ε

∣∣∣∣ε=0

(g1) + ∂φ(g1)g2

}IB(g1)

]

=∫B

{∂φ(ε)

∂ε

∣∣∣∣ε=0

(x) + ∂φ(ε)(x)E[g2|g1 = x]}n[x | 0,g1

]dx.

As for the expectation of φ(0)(g1)∂IB(g1)g2, we notice that φ(0)(g1)g2 ∈ D∞(R).

Then by using the integration by parts formula for Wiener functionals, we have

E[φ(0)(g1)∂IB(g1)g2

]= E[G(w)IB(g1)]= E

[E[G(w)|g1 = x]IB(g1)

]=∫B

E[G(w)|g1 = x]n[x | 0,g1

]dx

≡∫B

p1(x) dx

for a smooth Wiener functional G(w). In order to obtain an explicit representationof p1(x), we set Bx = (−∞, x] . Then we have

E[φ(0)(g1)∂IBx (g1)g2

]= ∫ ∞−∞

φ(0)(y)E[g2|g1 = y]∂IBx (y)n[y | 0,g1

]dy

= −∫ ∞−∞

φ(0)(y)E[g2|g1 = y]δx(y)n[y | 0,g1

]dy

= −φ(0)(x)E[g2|g1 = x]n[x | 0,g1

],

where δx(y) denotes the delta function of y at x. By differentiating the above term


with respect to x, we have

p1(x) = ∂

∂x

[−φ(0)(x)E[g2|g1 = x]n[x | 0,g1

]].

By adding two terms, we have the explicit formula for E[�1]. Third, we shallderive an explicit representation for E[�2], which is more complicated. For thispurpose, we write it as

E[�2] =∫B

p21(x) dx +∫B

p22(x) dx +∫B

p23(x) dx,

where p2i (i = 1,2,3) corresponds to each line of �2. The first term p21(x) canbe calculated directly as E[�1] by using the integration by parts formula and isgiven by

p21(x) = ∂

∂x

[−{

∂φ(ε)

∂ε

∣∣∣∣ε=0

(x)E[g2|g1 = x]

+ ∂φ(ε)(x)E[g2

2 |g1 = x]}

n[x | 0,g1

]].

For the second term, we only need the standard differentiation, and it is given by

p22(x) =[

1

2

∂2φ(ε)

∂ε2

∣∣∣∣ε=0

(x) +{

∂2φ(ε)(x)

∂x ∂ε

∣∣∣∣ε=0

}E[g2|g1 = x]

+ ∂φ(0)(x)E[g3|g1 = x] + 1

2∂2φ(0)(x)E

[g2

2 |g1 = x]]

n[x | 0,g1

].

In order to derive p23(x), first we need an expression of the second-ordergeneralized derivatives of Wiener functional E[1

2φ(0)(g1)∂2IB(g1)g

22]. By taking

B = Bx = (−∞, x] and using the integration by parts formulas for Wienerfunctionals, we have

E[

1

2φ(0)(g1)∂

2IBx (g1)g22

]

=∫ ∞−∞

∂2IBx (y)

{1

2φ(0)(y)E

[g2

2 |g1 = y]n[y | 0,g1

]}dy

= ∂

∂x

∫ ∞−∞

δx(y)

{1

2φ(0)(y)E

[g2

2 |g1 = y]n[y | 0,g1

]}dy

= ∂

∂x

{1

2φ(0)(x)E

[g2

2 |g1 = x]n[x | 0,g1

]}

=∫ x

−∞∂2

∂y2

{1

2φ(0)(y)E

[g2

2 |g1 = y]n[y | 0,g1

]}dy.


For the term of E[φ(0)(g1)∂IB(g1)g3], we obtain

E[φ(0)(g1)∂IBx g3

]= ∫ x

−∞∂

∂y

{−φ(0)(y)E[g3|g1 = y]n[y | 0,g1

]}dy.

Hence, p23(x) is given by

p23(x) = ∂2

∂x2

{1

2φ(0)(x)E

[g2

2 |g1 = x]n[x | 0,g1

]}

+ ∂

∂x

{−φ(0)(x)E[g3|g1 = x]n[x | 0,g1

]}.

Finally, by collecting and rearranging each term of p21(x),p22(x), and p23(x), wehave the result. �

If we take a particular function φ(ε)(x), we can derive the correspondingformulas in the asymptotic expansion. Here we give simple examples when d =m = 1 for the illustration. When we take φ(ε)(x) ≡ 1 and B = (−∞, x], then wehave an asymptotic expansion of the distribution function, which is given by

P({

X(ε)T ≤ x

})∼∫ x

−∞n[y | 0,g1

]dy

+ ε

∫ x

−∞

[− ∂

∂yE[g2|g1 = y]n[y | 0,g1

]]dy

+ ε2∫ x

−∞

[1

2

∂2

∂y2

{E[g2

2 |g1 = y]n[y | 0,g1

]}

+ ∂

∂y

{−E[g3|g1 = y]n[y | 0,g1

]}]dy + · · · .

Also, for the the pay-off function of European call options, we set φ(ε)(x) = x + y

for a constant y and B = [−y,∞). Then we have

E[(x + y)+

]∼ ∫ ∞−y

(y + x) n[x | 0,g1

]dx

+ ε

∫ ∞−y

x

[− ∂

∂xE[g2|g1 = x]n[x | 0,g1

]]dx

+ ε2∫ ∞−y

x

[∂

∂x

{−E[g3|g1 = x]n[x | 0,g1

]}

+ 1

2

∂2

∂x2

{E[g2

2 |g1 = x]n[x | 0,g1

]}]dx + · · · .

These results we have obtained are equivalent to the formulas for the Europeancall options previously reported by Takahashi (1999). In fact, the formulae in


Theorem 3.4 are equivalent to those obtained by the characteristic functionsand their the Fourier inversions originally obtained by Fujikoshi, Morimune,Kunitomo and Taniguchi (1982). They have been extensively used in Kunitomoand Takahashi (1995, 2001) and Takahashi (1999).

As a more complicated application, we consider the problem arising in thevaluation of the Asian options mainly because it illustrates a wide applicabilityof our approach in mathematical finance. The explicit formulas have been derivedin Section 3.2 of Takahashi (1999). The terminal pay-off for the Asian options isdependent on

A(ε)T =

∫ T

0f(S(ε)

s

)ds,(3.19)

where f (·) is a smooth function, which is in C∞(Rd → R) and S(ε)s satisfies

(3.1)–(3.3). In this case we take A(0)T = ∫ T

0 f (S(0)s ) ds and we need to derive the

asymptotic expansion of the random variable F(ε)T = (1/ε)[A(ε)

T − A(0)T ]. By using

a formal Taylor expansion, an asymptotic expansion of the random variable F(ε)T

can be written as

F(ε)T ∼ g1T (A) + εg2T (A) + ε2g3T (A) + · · · ,

where gnT (A) (n ≥ 1) are defined by gnT (n ≥ 1) in Theorem 3.1 as

gnT (A) =∫ T

0

n∑k=1

m1+···+mk=n

[1

k!d∑

l1,...,lk=1

∂kl1,...,lk

f(S(0)

s , s) k∏j=1

g(lj )mj ,s

]ds

and mj (j = 1, . . . , k) are positive integers.

By using the smoothness condition of f (·), S(ε)T ∈ D∞(Rd), and g1s, g2s ,

g3s , . . . ∈ D∞(Rd), we see that F(ε)T has an asymptotic expansion, which is in

D∞(R) as ε ↓ 0 with gkT (A) ∈ D∞(R) (k = 1,2, . . .). The Malliavin covarianceof F

(ε)T , which is denoted as σMC(F

(ε)T ), is given by

σMC(F

(ε)T

)=∫ T

0

∣∣∣∣{∫ T

u∂f(S(ε)

s

)Y (ε)

s ds

}Y (ε)−1

u σ(S(ε)

u , u)∣∣∣∣

2

du.(3.20)

If we define ηεc(A) as before by

ηεc(A) = c

∫ T

0

∣∣∣∣∫ T

u∂f(S(ε)

s

)Y (ε)

s dsY (ε)−1u σ

(S(ε)

u , u)

−∫ T

u∂f(S(0)

s

)Ys dsY−1

u σ(S(0)

u , u)∣∣∣∣

2

du,

then we have the corresponding results as for ηεc(A) instead of ηε

c exactly in thesame way. As before, we need to make use of Lemma A.2 in the Appendix.


Consequently, we can apply Theorem 2.1 to ψ(ηεc(A))φ(F

(ε)T )IB(F

(ε)T ), and the

same results as in Theorem 3.3 and Corollary 3.1 can be obtainable for F(ε)T if we

use the next assumption instead of Assumption I.

ASSUMPTION I′. For any T > 0,

g1(A) =∫ T

0

∣∣∣∣{∫ T

u∂f(S(0)

s

)Ys ds

}Y−1

u σ(S(0)

u , u)∣∣∣∣

2

du > 0.(3.21)

THEOREM 3.5. Under the assumptions in (3.1)–(3.3), the smoothness off (·) in C∞ and Assumption I′, ψ(ηε

c(A))φ(ε)(F(ε)T )IB(F

(ε)T ) has an asymptotic

expansion

ψ(ηε

c(A))φ(ε)

(F

(ε)T

)IB(F

(ε)T

)∼ �0 + ε�1 + · · ·(3.22)

in D−∞(R) as ε ↓ 0, where �0,�1, . . . are determined by the formal Taylorexpansion with respect to F

(ε)T (∼ g1T (A) + εg2T (A) + · · ·), and φ(ε)(·), ψ(·) and

IB(·) are defined as in Theorem 3.3.

COROLLARY 3.2. Under the assumptions in (3.1)–(3.3), the smoothness off (·) in C∞ and Assumption I′, an asymptotic expansion of E[φ(ε)(F

(ε)T )IB(F

(ε)T )]

is given by

E[φ(ε)

(F

(ε)T

)IB(F

(ε)T

)]∼ E[ψ(ηε

c(Z))φ(ε)(F (ε))IB(F (ε))

]∼ E[�0] + εE[�1] + · · ·

(3.23)

as ε ↓ 0, where φ(ε)(·), ψ(·), �j (j ≥ 0) and B are defined as in Theorem 3.3.

REMARK. The general valuation problem of financial contingent claimsincluding the European options and the Asian options in the Black–Scholeseconomy can be simply defined as finding its “fair” value at financial markets. LetV (T ) be the pay-off of a contingent claim at terminal period T . Then the standardmartingale theory in financial economics predicts that the fair price of V (T ) attime t (0 ≤ t < T ) should be given by

Vt(T ) = Et

[exp

[−∫ T

tr(S(ε)

v , v)dv

]V (T )

],

where Et [·] stands for the conditional expectation operator given the informationavailable at t with respect to the equivalent martingale measure Q. Then we canexpand the expected value with respect to the parameter ε and use the formulasin Theorem 3.4. Takahashi (1995, 1999) has already given many asymptoticexpansion formulas for the examples we have mentioned in this section includingthe Asian options and others when r is a positive constant. In this case, theconditions in (3.2) on the drift terms are automatically satisfied.


4. Validity in the term structure model of interest rates. Let (,F ,Q,

{Ft }t∈[0,T ]) be the filtered probability space with T < +∞ and {wit ; i = 1, . . . ,m}

are independent Brownian motions with respect to {Ft }. Let also T = {(s, t) | 0 ≤s ≤ t ≤ T } be a compact set in R2. We shall consider a class of randomfields f (ε)(s, t) : T → R which are adapted with respect to {Ft} and satisfy thestochastic integral equation

f (ε)(s, t) = f (0, t) + ε2∫ s

0

[m∑

i=1

σ i(f (ε)(v, t), v, t

)

×∫ t

vσ i(f (ε)(v, y), v, y

)dy

]dv

+ ε

∫ s

0

m∑i=1

σ i(f (ε)(v, t), v, t

)dwi

v.

(4.1)

We note that there are integrals with respect to the maturity parameter in the driftterm involving {σ i(f (ε)(v, y), v, y) (i = 1, . . . ,m), (v, y) ∈ T }. We can constructsuch integrals recursively by considering the discretized versions with respect tothe maturity argument as∫ t

vσ i(f (ε)

(v,ψn′(y)

), v,ψn′(y)

)dy

for 0 ≤ v ≤ s ≤ y ≤ t ≤ T , where ψn′(s) = (k + 1)T /2n′if s ∈ (kT /2n′

,

(k + 1)T /2n′ ] (k = 0, . . . ,2n′ − 1;n′ ≥ 1). Then we can make the solutionf (ε)(s,ψn′(t)) of (4.1) to be adapted with respect to Fs (0 ≤ s ≤ t ≤ T ). Actually,by using the standard argument in stochastic analysis, we can further discretize thevolatility functions as σ i(f (ε)(φn(v),ψn′(y)),φn(v),ψn′(y)) for 0 ≤ v ≤ y ≤ T ,where φn(v) = kT /2n if v ∈ [kT /2n, (k + 1)T /2n) (k = 0, . . . ,2n − 1; n ≥n′ ≥ 1). Then by using a real polynomial function p1(x1, x2, . . . , x2n), the firstpart of the solution of the discretized version can be written as

f n,n′(ε)(s, τ (n′))= p1

([h1](w), [h2](w), . . . ,[hφ∗

n(s)+1](w), ·)

for 0 ≤ s ≤ τ (n′), where τ (n′) = T/2n′and φ∗

n(s) = φn(s)/τ (n). Also byusing real polynomial functions pk(·) (k = 2, . . . ,2n′

), we have a recursiverepresentation as

f n,n′(ε)(s, kτ (n′))= pk

([h1](w), . . . ,[hφ∗

n(s)+1](w), ·,pk−1(·), . . . , p1(·))

for k = 2, . . . ,2n′and 0 ≤ s ≤ kτ (n′). Hence the solution of the discretized version

of (4.1) can be represented as polynomial functions of [h1](w), [h2](w), . . . ,

[h2n](w).Returning to (4.1), we make the following assumptions on the volatility

functions in this section.


ASSUMPTION II. The volatility functions σ i(f (ε)(s, t), s, t) (i = 1, . . . ,m)

are nonnegative, measurable, bounded, and smooth in the first argument, and allderivatives are bounded uniformly in ε. The initial forward rates f (0, t) are alsoLipschitz continuous with respect to t .

ASSUMPTION III. For any 0 < s ≤ t ≤ T ,

(s, t) =∫ s

0

m∑i=1

[σ (0)i(v, t)

]2dv > 0,(4.2)

where σ (0)i(v, t) = σ i(f (ε)(v, t), v, t)|ε=0.

The conditions stated in Assumption II exclude the possibility of explosions forthe solution of (4.1). Assumption III ensures the key condition of nondegeneracy ofthe Malliavin covariance in our problem, which is essential for the validityof the asymptotic expansion approach for the forward rate processes. UnderAssumption II we can get the stochastic expansions of the forward rates and spotinterest rates processes. The starting point of our discussion is a simplified versionof the result by Morton (1989) on the existence of the solution of the stochasticintegral equation (4.1) for forward rate processes.

THEOREM 4.1 [Morton (1989)]. Under Assumption II, there exists a jointlycontinuous process {f (ε)(s, t),0 ≤ s ≤ t ≤ T } satisfying (4.1) with ε = 1. Thereexists only one solution of (4.1) with ε = 1.

In the rest of this section we often refer to the case of m = 1 whenever wecan avoid complicated notation and the proofs are as if it is the general casewithout loss of generality. For that purpose we use the convention wv = w1

v

and σ(f (ε)(s, t), s, t) = σ 1(f (ε)(s, t), s, t) when m = 1. We construct the com-pletion of the polynomial functions of pk([h1](w), [h2](w), . . . , [h2n](w)) (k =1, . . . ,2n′

). With a fixed n′ we will abuse the notation slightly and denote the re-sulting totality of polynomials and the totality of smooth functionals as P (R) andS(R), respectively, in this section. If we denote the resulting forward processes asf n′(ε)(s,ψn′(t)), then for any p > 1 we have

E

[sup

0≤s≤t≤T

∣∣f (ε)(s, t) − f n′(ε)(s,ψn′(t))∣∣p]→ 0(4.3)

as n′ → +∞ by using the standard arguments in stochastic analysis. [See ChaptersIV and V of Ikeda and Watanabe (1989).] Hence in the rest of this section weconsider the sequence of {f n′(ε)(s,ψn′(t))} as if they were {f (ε)(s, t)} to avoidthe resulting tedious arguments. The kth order H -derivative (k ≥ 1) of the forwardrate process f n′(ε)(s,ψn′(t)) ∈ S(R) is denoted as Dkf (ε)(s, t) ∈ S(H⊗k ⊗ R).

We summarize the first major result in this section on the sequence of forwardprocesses {f (ε)(s, t)} as the next theorem. The proof is the result of lengthyarguments on the higher order H -derivatives of {f (ε)(s, t)} given in the Appendix.


THEOREM 4.2. Suppose Assumption II hold for the forward rate processesfollowing (4.1). Then for any ε ∈ (0,1] and (s, t) ∈ T , f (ε)(s, t) ∈ D∞(R).

Next we consider the asymptotic behavior of a functional,

F (ε)(s, t) = 1

ε

[f (ε)(s, t) − f (0)(0, t)

](4.4)

as ε → 0. Then the H -derivative of F (ε)(s, t) can be represented as

DhF(ε)(s, t) =

∫ s

0Y (ε)(s, t)Y (ε)−1(v, t)C(ε)(v, t) dv,(4.5)

where the stochastic process {Y (ε)(s, t),0 ≤ s ≤ t ≤ T } is defined as the solutionof the stochastic integral equations

Y (ε)(s, t) = 1 + ε2∫ s

0

[∂σ(f (ε)(v, t), v, t

)

×∫ t

vσ(f (ε)(v, y), v, y

)dy

]Y (ε)(v, t) dv(4.6)

+ ε

∫ s

0∂σ(f (ε)(v, t), v, t

)Y (ε)(v, t) dwv

and

C(ε)(v, t) = σ(f (ε)(v, t), v, t

)hv

+ εσ(f (ε)(v, t), v, t

) ∫ t

v∂σ(f (ε)(v, y), v, y

)Dhf

(ε)(v, y) dy.

We notice that the coefficients of Y (ε)(s, t) on the right-hand side of (4.6) arebounded under Assumption II. Hence for any 1 < p < +∞, 0 < ε ≤ 1, we haveE[|Y (ε)(s, t)|p]+E[|Y (ε)−1(s, t)|p] < +∞. The proof of this result has been givenin Kunitomo and Takahashi (2001). By rearranging terms in the integrands of (4.5),we have the representation

DhF(ε)(s, t) =

∫ s

0ν

(ε,1)s,t (u)hu du,(4.7)

where

ν(ε,1)s,t (u) = Y (ε)(s, t)Y (ε)−1(u, t)σ

(f (ε)(u, t), u, t

)+ ε

∫ s

uY (ε)(s, t)Y (ε)−1(v, t)σ

(f (ε)(v, t), v, t

)

×(∫ t

v∂σ(f (ε)(v, y), v, y

)ξ (ε,1)v,y (u) dy

)dv,


and {ξ (ε,1)v,y (u)} are defined by {ξ (1,1)

v,y (u)} of (A.8) in the Appendix by replacing(1,1) with (ε,1). Hence the Malliavin covariance of F (ε)(s, t), σMC(F (ε)(s, t)),

is obtained by

〈DF(ε),DF (ε)〉H =∫ s

0

∣∣ν(ε,1)s,t (u)

∣∣2 du.(4.8)

Let

η(ε)c (s, t) = c

∫ s

0

∣∣∣∣ε(∫ s

uY (ε)(s, t)Y (ε)−1(v, t)σ

(f (ε)(v, t), v, t

)

×∫ t

v∂σ(f (ε)(v, y), v, y

)ξ (ε,1)v,y (u) dy dv

)∣∣∣∣2

du

+ c

∫ s

0

∣∣Y (ε)(s, t)Y (ε)−1(u, t)σ(f (ε)(u, t), u, t

)− σ

(f (0)(u, t), u, t

)∣∣2 du,

for a positive constant c > 0. We notice that the condition in Assumption IIIis equivalent to the nondegeneracy condition of (4.8) because Y (0)(v, t) = 1 for(v, t) ∈ T . Again by using the similar arguments as Lemma A.2 in the Appendix,for (s, t) ∈ T and any k ≥ 1,

limε→0

ε−kP{|η(ε)

c (s, t)| > 12

}= 0.(4.9)

Then by a similar argument as Theorem 3.2 in Section 3, we shall obtain atruncated version of the nondegeneracy condition of the Malliavin covariance forthe spot interest rates and forward rates processes, which is the key step to showthe validity of the asymptotic expansion approach.

THEOREM 4.3. Under Assumptions II and III, the Malliavin covarianceσ(F (ε)(s, t)) of F (ε)(s, t) is uniformly nondegenerate in the sense that there existsc0 > 0 such that for any c > c0 and any p > 1,

sup0<ε≤1

E[I(|η(ε)

c | ≤ 1)(

σMC(F (ε)(s, t)

))−p]< +∞,(4.10)

where I (·) is the indicator function.

Hence the validity of the asymptotic expansions of the distribution function orthe density function of instantaneous spot rate, and forward rates can be obtainedunder Assumption II and Assumption III because we have proved that a set ofconditions in Theorem 2.1 are satisfied.

We now return to the general case where m ≥ 1. By expanding the Wienerfunctional F (ε)(s, t), we can write

F (ε)(s, t) ∼ A1(s, t) + εA2(s, t) + · · · .


The coefficients in the asymptotic expansion can be obtained by applying a formalTaylor expansion and An(s, t) (n ≥ 1) are given by

A1(s, t) =m∑

i=1

∫ s

0σ i(f (0)(0, t), v, t

)dwi

v,(4.11)

and for n ≥ 2,

An(s, t) =m∑

i′=1

∫ s

0

j1+···+jk=k+lj ′

1+···+j ′k′=k′+l′∑

k,k′,l,l′≥0k+k′+l+l′+2=n

1

k!k′!∂kσ i′(f (0)(v, t), v, t

) k∏i∗=1

Aji∗ (v, t)

×∫ t

v∂k′

σ i′(f (0)(v, y), v, y) k∏

i=1

Aji(v, y) dy

dv(4.12)

+m∑

i′=1

∫ s

0

j1+···+jk=k+l∑k,l≥0

k+l+1=n

1

k!∂kσ i′(f (0)(v, t), v, t

) k∏i∗=1

Aji∗ (v, t)

dwi′

v .

Some of these formulas have been previously obtained by Kunitomo and Takahashi(1995, 2001). By applying similar arguments, which are actually quite tedious, wecan show that the Lp-boundedness of any order H -derivatives of An(s, t) for any0 ≤ s ≤ t ≤ T and integers n ≥ 1. Then we conclude that An(s, t) ∈ D∞(R) forany n ≥ 1 and summarize the result as the next theorem.

THEOREM 4.4. Under Assumption II, F (ε)(s, t) is in D∞(R) and has anasymptotic expansion,

F (ε)(s, t) ∼ A1(s, t) + εA2(s, t) + · · ·(4.13)

as ε ↓ 0 and A1(s, t),A2(s, t), . . . ∈ D∞(R).

We notice that the Malliavin covariance is nondegenerate because (s, t) isnondegenerate, which is in turn the variance of A1(s, t). Then we have theGaussian random variable as the leading term in (4.13) and we can use the samemethod as in Section 3 to derive the asymptotic expansion of the expected values ofrandom variables. By applying the corresponding one as Theorem 2.1 for D∞(R),it has a proper asymptotic expansion as ε → 0 in D

−∞(R). Hence we obtain the

next result.

THEOREM 4.5. Under Assumptions II and III, an asymptotic expansion ofE[φ(ε)(F (ε))IB(F (ε))] is given by

E[φ(ε)(F (ε))IB(F (ε))

]∼ E[ψ(ηε

c)φ(ε)(F (ε))IB(F (ε))

]∼ E[�0] + εE[�1] + · · ·(4.14)


as ε ↓ 0, where �j (j ≥ 0) are obtained by a formal Taylor expansion of the left-hand side in the expectation operator with respect to F (ε)(s, t) and ψ(ηε

c), φ(ε)(·),and IB(·) are defined as in Theorem 3.3.

Also it is straightforward to obtain the similar nondegeneracy conditions ofthe Malliavin covariance for the discounted coupon bond price process and theaverage interest rate process. We note that the discount bond price process is givenby (1.5). Then using (4.1) and Itô’s lemma, we can consider the stochastic processG(ε)(t, T ,p) = [P (ε)(t, T )]p for any integer p > 1, which is the solution of thestochastic integral equation

G(ε)(t, T ,p)

= G(ε)(0, T ,p)

+∫ t

0

[pr(ε)(v)

+ p2 − p

2ε2

m∑i=1

(∫ T

tσ i(f (ε)(v, y), v, y

)dy

)2]G(ε)(v, T ,p)dv

+m∑

i=1

∫ t

0(−pε)

[∫ T

tσ i(f (ε)(v, y), v, y

)dy

]G(ε)(v, T ,p)dwi

v.

Hence by using the fact that E[|r(ε)(t)|p] < +∞ for any p > 1 and 0 ≤ s ≤ t ≤ T

under Assumption II, we have E[|P (ε)(s, t)|p] < +∞. Then we can investigate theproperties of the H -derivatives on the set of discount bond price processes as forthe forward rate and spot rate processes we have discussed. Because the essentialarguments are the same, we only report the result.

THEOREM 4.6. Under Assumption II for the forward rate processes, for anyε ∈ (0,1] and 0 ≤ t ≤ T , P (ε)(t, T ) is in D∞(R) and has an asymptotic expansion

P (ε)(t, T ) ∼ P (t, T ) + εB1(t, T ) + ε2B2(t, T ) + · · ·(4.15)

as ε ↓ 0 and B1(t, T ),B2(t, T ), . . . ∈ D∞(R), where P (t, T ) (= P (0)(t, T )),Bj (t, T ) (j ≥ 1) are obtained by a formal Taylor expansion of P (ε)(t, T ) through(1.5), (4.4) and (4.13).

More generally, the valuation problem of many interest rates based contingentclaims in the complete market can be simply defined as to find the “fair” value of afunction of a series of bond prices at financial markets. Then the fair price of V (T )

at time t (0 ≤ t < T ) should be given by

Vt(T ) = Et

[exp

[−∫ T

tr(ε)(s) ds

]V (T )

],


where V (T ) is the pay-off of a contingent claim at the terminal period T and Et [ · ]stands for the conditional expectation operator given the information available at t

with respect to the equivalent martingale measure Q. Because we can derive anasymptotic expansion of the spot interest rate r(ε)(s), it is straightforward to obtainthe fair value of interest rates-based contingent claims.

For instance, most interest rates-based contingent claims can be regarded asfunctionals of bond prices with different maturities. Let {cj , j = 1, . . . , k} be asequence of nonnegative coupon payments and {Tj , j = 1, . . . , k} be a sequenceof payment periods satisfying the condition 0 ≤ t < T1 ≤ · · · ≤ Tk ≤ T ≤ +∞.

Then the price of the coupon bond with coupon payments {cj , j = 1, . . . , k} at t isgiven by

P(ε)k,{Tj },{cj }(t) =

k∑j=1

cjP(ε)(t, Tj ),(4.16)

where {P (ε)(t, Tj ), j = 1, . . . , k} are the prices of zero coupon bonds withdifferent maturities. The normalized random variable for the call options on thediscounted coupon bond at the initial period t = 0 is given by

R(ε)k,{Tj },{cj }(t)

= 1

ε

{exp

[−∫ t

0r(ε)(s) ds

][P

(ε)k,{Tj },{cj }(t) − K

]

−[

k∑j=1

cjP (0, Tj ) − KP(0, t)

]},

where 0 < t < T1 ≤ · · · ≤ Tk and K is a fixed real constant. This random variablehas an intuitive interpretation in financial applications. Its meaning and the relatedadditional assumptions for practical applications have been discussed in Section 3of Kunitomo and Takahashi (2001). By using (1.5) and (4.16), the first orderH -derivative of R

(ε)k,{Tj },{cj }(t) can be represented as

Dh

[R

(ε)k,{Tj },{cj }(t)

]

= − exp[−∫ t

0r(ε)(s) ds

][P

(ε)k,{Tj },{cj }(t) − K

] ∫ t

0Dh

[F (ε)(s, s)

]ds

− exp[−∫ t

0r(ε)(s) ds

][ k∑j=1

cjP(ε)(t, Tj )

∫ Tj

TDh

[F (ε)(t, u)

]du

],

where F (ε)(t, u) is defined by (4.4).From this expression we can obtain a simplified representation of the first-order

H -derivative in this case as before. Hence we can obtain the asymptotic expansion


of the expected pay-off value of a coupon bond if we use the condition ensuringthe nondegeneracy of the Malliavin covariance. The proof of the next theorem issimilar to those in the previous results.

ASSUMPTION IV. For any 0 < t < T1 ≤ · · · ≤ Tk ,

g1(k) =∫ t

0σ ∗

g1(v)σ ∗′

g1(v) dv > 0,(4.17)

where σ ∗g1

(v) = −[∑kj=1 cjP (0, Tj ) − KP(0, t)]σ (0)

t (v) − ∑ki=1 cjP (0, Tj ) ×

σ(0)t,Tj

(v), and σ(0)t (v) and σ

(0)t,Tj

(v) are 1 × m vectors such that σ(0)t (v) =

[∫ tv σ

(0)i (v, y) dy] and σ

(0)t,Tj

(v) = [∫ Tj

t σ(0)i (v, u) du].

THEOREM 4.7. Under Assumptions II and IV, an asymptotic expansion ofE[φ(ε)(R

(ε)k,{Tj },{cj }(t))IB(R

(ε)k,{Tj },{cj }(t))] is given by

E[φ(ε)

(R

(ε)k,{Tj },{cj }(t)

)IB(R

(ε)k,{Tj },{cj }(t)

)]∼ E

[ψ(ηε

c)φ(ε)(R(ε)

k,{Tj },{cj }(t))IB(R

(ε)k,{Tj },{cj }(t)

)](4.18)

∼ E[�∗0] + εE[�∗

1] + · · ·as ε ↓ 0, where �∗

j (j ≥ 0) are obtained by a formal Taylor expansion of the left-

hand side in the expectation operator, and ψ(ηεc), φ(ε)(·) and IB(·) are defined as

in Theorem 3.3.

We briefly mention two examples of interest rates-based contingent claims.The pay-off function of the call bond options with coupon payments {cj , j =1, . . . , k} at {Tj , j = 1, . . . , k} can be written as V (1)(T ) = [P (ε)

k,{Tj },{cj }(T )−K]+,

where K is a fixed strike price. In this case we can take φ(ε)(x) = x + y fora constant y and B = [−y,∞). As another example we should mention thepay-off function of the call options on average interest rates, which is givenby V (2)(T ) = [ 1

T

∫ T0 Lτ(t) dt − K]+, where the yield of a zero coupon bond

at t with time to maturity of τ (0 < t < t + τ < Tk) years is given by Lτ(t) =[1/P (ε)(t, t + τ ) − 1]/τ . Then we can apply the asymptotic expansion methodwith some additional assumptions. For these two examples and others, Kunitomoand Takahashi (1995, 2001) have derived more explicit formulas of the asymptoticexpansions in details.

REMARK. We should mention that we can use the equivalence between theformulas by the expected values of the generalized Wiener functionals and thosederived by using the simple inversion technique for the characteristic functions ofrandom variables as we have discussed in Section 3.


5. Numerical examples. In this section we will present numerical examplesin order to illustrate the usefulness of approximations obtained by the asymptoticexpansion method we have discussed. There have been many examples andsome of them have been already reported by Kunitomo and Takahashi (1995,2001) and Takahashi (1999). As an example of the Black–Scholes economy,we give some numerical results on the average call options for the squareroot process and the log-normal process for the underlying asset prices. Underthe equivalent martingale measure, we assume that the processes of the one-dimensional underlying asset are given by

dS1(ε1) = (r − q)S1(ε1) dt + ε1σ(S1(ε1))1/2 dwt,(5.1)

dS2(ε2) = (r − q)S2(ε2) dt + ε2S2(ε2) dwt,(5.2)

where ε1, ε2, σ are parameters, and r and q denote the risk-free interest rateand a dividend yield (we assume both are positive constants), respectively, andwt denotes the one-dimensional Brownian motion. The theoretical value of theaverage (or Asian) call options at time 0 should be given by

E0

[exp(−rT )max

{1

T

∫ T

0Si(εi )

u du − K,0}]

, i = 1,2,(5.3)

where K is the strike price. The terminal pay-off in this example is a special caseof (3.19) and then we can apply Corollary 3.2 to this case.

NOTE. In this example the volatility function is not smooth at the origin andwe need to use a smoothed version of the square root process at the origin forthe mathmatical point of view. It is possible to show that the smoothing and thetruncation by a large threshold value do not make significant differences and theeffects are negligible in the small disturbance asymptotic theory.

In Tables 1–5 we have calculated the differences between the Monte Carlo valueand the second-order approximations based on asymptotic expansions. The valuesin Tables 4 and 5 are calculated in terms of the basis points except the percentagepoints (%). Difference (bp) and difference rate (%) are calculated by the deviationsfrom the Monte Carlo results in (1). The values in the last columns were calculatedby setting ε1 = ε2 = 0.0 (or ε = 0.0), which could be regarded as the zeroth orderapproximations.

Table 1 shows the numerical values of the average call options for the squareroot processes of the underlying asset which represents an equity index withno dividend. We take the spot price S0 = 40.00, the risk-free rate r = 5%, theparameter σ = 10.00 and the six month maturity date. The parameters ε1 and ε2were set so that the instantaneous volatility is equivalent to the correspondingvolatility of the 30% log-normal process (i.e., ε1 = 0.189737 and ε2 = 0.3).


TABLE 1Average call options on equity square root processes

Strike price 45 40 35

(1) Monte Carlo 0.5221 2.1758 5.6468(2) Stochastic expansion 0.5228 2.1788 5.6516Difference 0.00078 0.00301 0.00482Difference rate, % 0.15 0.14 0.09Value when ε1 = 0.0 0.0 0.4917 5.3683

Tables 2 and 3 are the numerical values of the average call options on the foreignexchange rate example when the underlying assets follow square root processesand the log-normal process, respectively. In this example we take the spot priceS0 = 100.00 and regard r as the risk-free interest rate in Japan and q as the risk-free interest rate in the U.S., and we set 3% and 5%, respectively. In Tables 2 and 3the spot price, the six month maturity date and the parameters ε1 and ε2 were setso that the instantaneous variance at time 0 are equivalent to 10% volatility of thelog-normal process (i.e., ε1 = 0.158114 and ε2 = 0.1).

For the purpose of comparison, the values by the Monte Carlo simulationsare also given, which are based on 500,000 trials implemented in each case.We also report the value based on the PDE numerical method developedby He and Takahashi (2000) for Table 3. The approximations given by theasymptotic expansion are those from the approximations up to the second orderfor Tables 1–3 where they are based on the total approximations consisting of thebasic deterministic terms, the first-order terms based on the Gaussian distributionand the additional second-order terms based on the non-Gaussian adjustments. InTables 1–3 it is apparent that we have enough accuracy of approximations forfinancial applications by the asymptotic expansion approach. More details of thisexample and other examples in the Black–Scholes economy have been discussedby Takahashi (1999).

TABLE 2Average call options on FX square root processes


(1) Monte Carlo 0.1721 1.3625 4.6858(2) Stochastic expansion 0.1730 1.3654 4.6931Difference 0.00090 0.00286 0.00730Difference rate, % 0.52 0.21 0.16Value when ε1 = 0.0 0.0 0.0 4.4346


TABLE 3Average options on FX log-normal process


(1) Monte Carlo simulation method 0.1840 1.3682 4.6793(2) Stochastic expansion 0.1830 1.3660 4.6800Difference rate, % −0.54 −0.16 0.01(3) Finite difference (Crank–Nicholson method) 0.1831 1.3656 4.6788Difference rate, % −0.49 −0.19 −0.01Value when ε2 = 0.0 0.0 0.0 4.4346

NOTE. Since the final formulas in our approximations are analytical whichare simple functions of the Gaussian distribution functions and some low-orderHermitian polynomials, the computation running times are negligible by anycomputational standards. Also at the suggestion of a referee we have added thedeterministic values in the last column by setting ε1 = ε2 = 0.0 (ε = 0.0 for theinterest rate based contingent claims), which could be regarded as the zeroth orderapproximations.

As an example of the non-Markovian term structure model of interest rates, wegive an example of swaptions in the HJM term structure model. For simplicity ofexposition, we assume that the instantaneous forward rate processes {f (ε)(s, t)}have one-factor volatility function as σ(f (ε)(s, t), s, t) = [f (ε)(s, t)]β, where0 ≤ β ≤ 1 and m = 1 in (4.1).

NOTE. We have used the truncated version of this forward rate processwhen 0 < β ≤ 1 because the original process theoretically could have explosivesolutions. For the Gaussian forward rate case other numerical valuation methodshave been known, but we report the results for comparative purposes. SeeKunitomo and Takahashi (2001) for the details.

Tables 4 and 5 show the numerical values of the call options of a swap contract(the swaption) for the case when β = 0 and ε = 0.01 (100 bp) and the case when

TABLE 4Swaption (Gaussian case)

Fixed rate % 7.18 6.16 5.13 4.10 3.08

(1) Monte Carlo 774.6 518.2 315.0 170.9 81.3(2) Stochastic expansion 774.8 518.5 315.1 171.15 81.2Difference (bp) 0.28 0.36 0.16 0.36 0.03Difference rate, % 0.04 0.07 0.05 0.21 0.04Value when ε = 0.0 689.1 344.5 0.0 0.0 0.0


TABLE 5Swaption (log-normal case)

Fixed rate % 7.18 6.16 5.13 4.10 3.08

(1) Monte Carlo 814.1 542.6 312.3 140.2 39.6(2) Stochastic expansion 819.1 546.5 315.1 143.3 42.3Difference (bp) 5.0 3.9 2.8 3.1 2.7Difference rate, % 0.6 0.7 0.9 2.2 6.8Value when ε = 0.0 689.1 344.5 0.0 0.0 0.0

β = 1 and ε = 0.2 (20%), respectively. In both cases we consider that the term ofthe underlying interest swap is five years, the time to expiration is also five years,and we set τ = 1 (year), T = 5, T1 = 6, . . . , T5 = 10 and k = 5. The present termstructure at t = 0 is assumed to be flat at 5% per year and we took cj = Sτ (j =1, . . . ,4), c5 = 1+Sτ , S = [P (0, T ) − P (0, T5)]/τ ∑5

j=1 P (0, Tj ) = 0.05171 andK = 1.00. In this example the theoretical value of swaption at time 0 should begiven by

E0

[exp

[−∫ T

0r(ε)(s) ds

]max

{k∑

j=1

cjP(ε)(T , Tj ) − K,0

}].(5.4)

Then we can apply Theorem 4.7 to this case. We have used the approximationsbased on the asymptotic expansions and examine their accuracy by Monte Carloresults for all cases. We have given the numerical results for the out-of-the-moneycase (S = 5.171% × 0.8,5.171% × 0.6), at-the-money case (S = 5.171%), andin-the-money case (S = 5.171% × 1.2,5.171% × 1.4). From Tables 4 and 5 wefind that the differences in the option values by the asymptotic expansion approachfor the Gaussian forward rates case are very small, and the differences of theoption values between the approximations and the Monte Carlo results for thegeometric Brownian forward rates case become slightly larger due to the non-Gaussianity of the underlying forward rates and the spot rates. Nonetheless, westill have enough acuracy in our approximations for financial applications sincethe differences between the approximations and the corresponding Monte Carloresults are usually within 3 bp in most cases. Kunitomo and Takahashi (2001) havediscussed more examples in the HJM term structure of the interest rates model.

6. Concluding remarks. This paper gives the mathematical validity of theasymptotic expansion approach for the valuation problem of financial contingentclaims when the underlying forward rates follow a general class of continuous Itôprocesses in the HJM term structure of the interest rates model and the underlyingasset prices follow a general class of diffusion processes in the Black–Scholeseconomy. Our method, called the small disturbance asymptotic theory, can beapplicable to a wide range of valuation problems of financial contingent claims.


Some of them have been discussed by Kunitomo and Takahashi (1995, 2001) andTakahashi (1999).

Since the asymptotic expansion approach can be justified rigorously by theWatanabe–Yoshida theory on the Malliavin calculus in stochastic analysis, it isnot an ad hoc method to give numerical approximations. In Section 5 of this paperand in our previous papers [Kunitomo and Takahashi (1995, 2001) and Takahashi(1999)], we have illustrated that the approximations we have obtained via theasymptotic expansion method can be satisfactory in many cases for practicalpurposes as well.

APPENDIX

In this Appendix we give some mathematical details omitted in Sections 3 and 4.First we present two inequalities which are useful to show that the truncation bythe random variable ηε

c of (3.13) in Section 3 is negligible in probability under theassumptions of (3.1)–(3.4) when we derive the asymptotic expansion of randomvariables.

NOTE. The present proof of Lemma A.1, which is simpler than our originalone, is due to the referee. Actually we only need the conditions given by (3.2)–(3.3)with k = 1 for Lemma A.1.

LEMMA A.1. There exist positive constants ai (i = 1,2) independent of ε

such that

P

(sup

0≤s≤T

[|S(ε)s − S(0)

s | + |Y (ε)s − Ys |]> a0

)≤ a1 exp(−a2ε

−2)(A.1)

for all a0 > 0, where S(ε)t and Y

(ε)t are defined by (3.1) and (3.11), respectively.

PROOF. Let Z(ε)t = (S

(ε)1t , . . . , S

(ε)dt , Y

(ε)11t , . . . , Y

(ε)ddt )

′be a d1 × 1 state

vector with d1 = d(d + 1). By using (3.1) for S(ε)t and (3.11) for Y

(ε)t , Z

(ε)t [=

(Z(ε)it )] follows the stochastic differential equation in the form of

Z(ε)iT = Zi

0 +∫ T

0bi(Z(ε)i

s , s)ds + ε

m∑j=1

∫ T

0ωij

(Z(ε)

s , s)dwj

s ,

where b(Z(ε)s , s) = (bi(Z

(ε)s , s)) and ω(Z

(ε)s , s) = (ωij (Z

(ε)s , s)) are Rd1 ×

[0, T ] → Rd1 and Rd1 × [0, T ] → Rd1 ⊗ Rm Borel measurable functions whichare smooth with respect to Z

(ε)s . By using the Lipschitz continuity, there exists a

positive constant K3 such that

∣∣Z(ε)t − Z

(0)t

∣∣≤ K3

∫ t

0

∣∣Z(ε)s − Z(0)

s

∣∣ds + sup0≤s≤t

∣∣∣∣∫ s

0εω(Z(ε)

u , u)dwu

∣∣∣∣.


Furthermore, by using the Gronwall inequality,

sup0≤s≤T

∣∣Z(ε)s − Z(0)

s

∣∣≤ sup0≤s≤T

∣∣∣∣∫ s

0ω(Z(ε)

u , u)dwu

∣∣∣∣εeK3T .

If we can assume that for the d1 × 1 vector θ there exists A (> 0) such that

sup|θ |=1,0≤t≤T

⟨θ,ω

(Z

(ε)t , t

)ω∗(Z(ε)

t , t)θ⟩≤ A < ∞,

we can apply the standard large deviation result given by Theorem 4.2.1 in Stroockand Varadhan (1979). Hence for any a0 > 0 we have

P

({sup

0≤t≤T

∣∣Z(ε)t − Z

(0)t

∣∣> a0

})

≤ P

({sup

0≤t≤T

∣∣∣∣∫ T

0ω(Z

(ε)t , t

)dwt

∣∣∣∣> a0

εeK3T

})

≤ 2d1 exp{−a2

0e−2K3T

2Ad1Tε−2

}.

When A is not bounded, let a stopping time be τ = inf0≤t≤T {|Z(ε)t − Z

(0)t | > a0}

for any a0 > 0. Then

P

({sup

0≤t≤T

∣∣Z(ε)t − Z

(0)t

∣∣> a0

})

≤ P

({τ < T, sup

0≤t≤τ

∣∣∣∣∫ T

0ω(Z

(ε)t , t

)dwt

∣∣∣∣> a0

εeK3T

})

≤ 2d1 exp(−a2

0e−2K3T

2Ad1Tε−2

),

where

sup|θ |=1,|Z(ε)

t −Z(0)t |≤a0

⟨θ,ω

(Z

(ε)t , t

)ω∗(Z(ε)

t , t)θ⟩≤ A < ∞.

�

LEMMA A.2. Let the random variable ηεc be defined by (3.13). Then for any

c > 0 ηεc is O(1) in D∞ (R) and for c0 > 0 there exist some positive constants

ci (i = 2,3,4), such that

P ({|ηεc | > c0}) ≤ c2 exp(−c3ε

−c4).(A.3)

PROOF. We notice that

|ηεc | = c

∫ T

0

∣∣Y (ε)T Y (ε)−1

s σ(S(ε)

s , s)− YT Y−1

s σ(S(0)

s , s)∣∣2 ds

≤ cT sup0≤s≤T

∣∣Y (ε)T Y (ε)−1

s σ(S(ε)

s , s)− YT Y−1

s σ(S(0)

s , s)∣∣2.


The set {|ηεc | > c0} is included in {sup0≤s≤T |Y (ε)

T Y(ε)−1s σ (S

(ε)s , s) − YT Y−1

s ×σ(S

(0)s , s)| > (c0/cT )1/2}. By setting a constant c1 = √

c0/(9cT ), we have

{|ηεc | > c0} ⊂

{sup

0≤s≤T

|YT Y−1s |∣∣σ (S(ε), s

)− σ(S(0)

s , s)∣∣> c1

}

∪{

sup0≤s≤T

|Y (ε)−1s |∣∣σ (S(ε)

s , s)∣∣∣∣Y (ε)

T − YT

∣∣> c1

}

∪{

sup0≤s≤T

|YT |∣∣σ (S(ε)s , s

)∣∣∣∣Y (ε)−1s − Y−1

s

∣∣> c1

}.

Then by using Lemma A.1 because of the boundedness of |YT Y−1s | and the

smoothness of σ(S(ε)s , s), we have the result. �

Next we shall give the proof of Theorem 4.2 in Section 4 by using three lemmasand some additional derivations.

PROOF OF THEOREM 4.2. Without loss of generality we only give the proofwhen m = 1. First we consider first-order derivatives {Df (1)(s, t)}. For any h ∈ H ,we successively define a sequence of random variables {ξ (l)(s, t); (s, t) ∈ T ,l ≥ 1} by the integral equation

ξ (l+1)(s, t) =∫ s

0

[∂σ(f (1)(v, t), v, t

) ∫ t

vσ(f (1)(v, y), v, y

)dy ξ(l)(v, t)

]dv

+∫ s

0

[σ(f (1)(v, t), v, t

) ∫ t

v∂σ(f (1)(v, y), v, y

)ξ (l)(v, y) dy

]dv

+∫ s

0∂σ(f (1)(v, t), v, t

)ξ (l)(v, t) dwv

+∫ s

0σ(f (1)(v, t), v, t

)hv dv,

(A.4)

where the initial condition is given by ξ (0)(s, t) = 0. Then we have the nextresult by using the standard method of stochastic analysis. The proof is given inKunitomo and Takahashi (1995).

LEMMA A.3. For any p > 1 and 0 ≤ s ≤ t ≤ T , E[|ξ (l)(s, t)|p] < ∞ (l > 1),and there exists a positive constant M4 such that

E[

sup0≤u≤s

∣∣ξ (l+1)(u, t) − ξ (l)(u, t)∣∣2]≤ 1

(l + 1)! [M4(t + 1)s]l+1.(A.5)

As l → ∞,

sup0≤s≤t≤T

E[

sup0≤u≤s

∣∣ξ (l+1)(u, t) − ξ (l)(u, t)∣∣2]→ 0.(A.6)


Using Lemma A.3 and the Chebyshev inequality, we have∞∑l=1

P

{sup

0≤u≤s≤t

∣∣ξ (l+1)(u, s) − ξ (l)(u, s)∣∣> 1

2l

}≤

∞∑l=1

1

l! [4M4(T + 1)T ]l < +∞.

Then by the Borel–Cantelli lemma, the sequence of random variables {ξ (l)(s, t)}converges uniformly on (s, t) ∈ T . Hence we have established the existence of theH -derivative of f (1)(s, t), which is given by the solution of the stochastic integralequation

Dhf(1)(s, t) =

∫ s

0

[∂σ(f (1)(v, t), v, t

)×∫ t

vσ(f (1)(v, y), v, y

)dy Dhf

(1)(v, t)

]dv

+∫ s

0

[σ(f (1)(v, t), v, t

)

×∫ t

v∂σ(f (1)(v, y), v, y

)Dhf

(1)(v, y) dy

]dv

+∫ s

0∂σ(f (1)(v, t), v, t

)Dhf

(1)(v, t) dwv

+∫ s

0σ(f (1)(v, t), v, t

)hv dv.

(A.7)

We note that for the spot rate process {r(ε)(t)} the H -derivative can be well definedby

Dhr(ε)(t) = lim

s→tDhf

(ε)(s, t).

We consider the random variables {ξ (1,1)s,t (u)} for (s, t) ∈ T (s, t) and 0 ≤ u ≤ s ≤

t ≤ T satisfying the stochastic integral equation when ε = 1,

ξ(ε,1)s,t (u) = ε2

∫ s

u∂σ(f (ε)(v, t), v, t

)ξ

(ε,1)v,t (u)

×∫ t

vσ(f (ε)(v, y), v, y

)dy dv

+ ε2∫ s

uσ(f (ε)(v, t), v, t

)

×∫ t

v∂σ(f (ε)(v, y), v, y

)ξ (ε,1)v,y (u) dy dv

+ ε

∫ s

u∂σ(f (ε)(v, t), v, t

)ξ

(ε,1)v,t (u) dwv

+ σ(f (ε)(u, t), u, t

).

(A.8)


Then we can show that ∫ s

0ξ

(1,1)s,t (u)hu du = Dhf

(1)(s, t).

In order to examine the existence of moments of {ξ (1,1)s,t (u)} and other related

random variables, we need the following inequality, whose proof is given inKunitomo and Takahashi (2001).

LEMMA A.4. Suppose for k0 > 0, k1 > 0, AN > 0 and 0 < s ≤ t ≤ T ,a function wN(u, s, t) satisfies (i) 0 < wN(u, s, t) ≤ AN and (ii):

wN(u, s, t) ≤ k0 + k1

[∫ s

uwN(u, v, t) dv +

∫ s

u

∫ t

vwN(u, v, y) dy dv

].(A.10)

Then

wN(u, s, t) ≤ k0ek1(1+t)s .(A.11)

As an illustration of our method based on Lemma A.4, we consider the truncatedrandom variable

ζNs,t (u) = [

ξ(1,1)s,t (u)

]IN(s, t),(A.12)

where IN(s, t) = 1 if sup0≤v≤s,v≤y≤t |ξv,y(u)| ≤ N and IN(s, t) = 0 otherwise. Byusing the boundedness conditions in Assumption II and hs being square integrable,we can show that there exist positive constants Mi (i = 5, . . . ,8) such that

∣∣ζNs,t (u)

∣∣p ≤ M5

∫ s

u

∣∣ζNv,t (u)

∣∣p dv + M6

∣∣∣∣∫ s

uζNv,t (u) dwv

∣∣∣∣p

(A.13)

+ M7

∫ s

u

∫ t

v

∣∣ζNv,y(u)

∣∣p dy dv + M8∣∣σ (f (1)(u, t), u, t

)∣∣p.

By using the martingale inequality, the expectation of the second term is lessthan M6E[∫ s

u |ζNv,t (u)|p dv]. Also the last term in (A.13) is bounded because

σ(·) is bounded. If we set wN(u, s, t) = E[|ζs,t (u)|p], then we can directlyapply Lemma A.4. By taking the limit of the expectation function wN(u, s, t) asN → ∞, we have E[|ξ (1,1)

s,t (u)|p] < +∞. By using similar arguments, we have theexistence of moments as summarized in the next lemma.

LEMMA A.5. Under Assumption II, for any p > 1 and 0 ≤ u ≤ s ≤ t ≤ T , wehave E[sup0≤u≤s |ξ (1,1)

s,t (u)|p] < +∞ and E[sup0≤u≤s |f (1)(u, t)|p] < ∞.

Using Lemma A.5 and the equivalence of two norms stated in Section 2.1, wenow have established the following property of the first-order H -derivative,

f (1)(s, t) ∈ ⋂1<p<+∞

D1p(R).


Since we have completed the investigation of the first-order H -derivative, ournext task is to investigate some properties of the higher order H -derivativesof f (1)(s, t). We shall use the induction argument and assume that f (1)(s, t) ∈⋂

1<p<+∞ Dkp(R) (k ≥ 1). Then we have

Dh[Dkf (1)(s, t)]=∫ s

0

[∂σ(f (1)(v, t), v, t

) ∫ t

vσ(f (1)(v, y), v, y

)dyDh

[Dkf (1)(v, t)

]]dv

+∫ s

0

[σ(f (1)(v, t), v, t

) ∫ t

v∂σ(f (1)(v, y), v, y

)

× Dh

[Dkf (1)(v, y)

]dy

]dv

+∫ s

0∂σ(f (1)(v, t), v, t

)Dh

[Dkf (1)(v, t)

]dw(v)

+∫ s

0

[G

(k)1

(∂lσ

(f (1)(v, t), v, t

),

Dlf (1)(v, t),Dhf (1)(v, t), v, t; l = 0, . . . , k)

×∫ t

vH

(k)1

(∂lσ

(f (1)(v, y), v, y

),

Dlf (1)(v, y),Dhf(1)(v, y), v, y; l = 0, . . . , k

)dy

]dv

+∫ s

0

[G

(k)2

(∂lσ

(f (1)(v, t), v, t

),

Dlf (1)(v, t),Dhf(1)(v, t), v, t; l = 0, . . . , k

)]dwv

+∫ s

0

[G

(k)3

(∂lσ

(f (1)(v, t), v, t

),

Dlf (1)(v, t),Dhf(1)(v, t), v, t; l = 0, . . . , k

)]dv

+∫ s

0

[G

(k)4

(∂lσ

(f (1)(v, t), v, t

),

Dlf (1)(v, t),Dhf(1)(v, t), v, t; l = 0, . . . , k

)]hv dv,

where H(k)1 (·) and G

(k)j (·) (j = 1, . . . ,4) are defined recursively and they are

actually a sequence of polynomial functions. Although the above stochasticintegral equation has many terms, the basic structure is the same as the first-order


H -derivative of f (1)(s, t). Now we define the random variables {ξ (1,k)s,t (u) (k ≥ 1)}

for 0 ≤ u ≤ s ≤ t ≤ T by

ξ(1,k+1)s,t (u)

=∫ s

0

[∂σ(f (1)(v, t), v, t

) ∫ t

vσ(f (1)(v, y), v, y

)dy ξ

(1,k+1)v,t (u)

]dv

+∫ s

0

[σ(f (1)(v, t), v, t

) ∫ t

v∂σ(f (1)(v, y), v, y

)ξ (1,k+1)v,y (u) dy

]dv

+∫ s

0∂σ(f (1)(v, t), v, t

)ξ

(1,k+1)v,t (u) dwv

+∫ s

0

[G

(k)1

(∂lσ

(f (1)(v, t), v, t

),

Dlf (1)(v, t), ξ(1,l)v,t (u), v, t; l = 0, . . . , k

)

×∫ t

vH

(k)1

(∂lσ

(f (1)(v, y), v, y

),

Dlf (1)(v, y), ξ (1,l)v,y (u), v, y; l = 0, . . . , k

)dy

]dv

+∫ s

0

[G

(k)2

(∂lσ

(f (1)(v, t), v, t

),

Dlf (1)(v, t), ξ(1,l)v,t (u), v, t; l = 0, . . . , k

)]dwv

+∫ s

0

[G

(k)3

(∂lσ

(f (1)(v, t), v, t

),

Dlf (1)(v, t), ξ(1,l)v,t (u), v, t; l = 0, . . . , k

)]dv

+∫ s

0

[G

(k)4

(∂lσ

(f (1)(v, t), v, t

),

Dlf (1)(v, t), ξ(1,l)v,t (u), v, t; l = 0, . . . , k

)]dv.

Then we can show that∫ s

0ξ

(1,k+1)s,t (u)hu du = Dh

[Dkf (1)(s, t)

].

From the above representation, we have

∣∣Dk+1f (1)(s, t)∣∣2H⊗(k+1) =

∫ s

0

∣∣ξ (1,k+1)s,t (u)

∣∣2H⊗k du.(A.14)


Applying Lemma A.4, repeating the procedure as Lemma A.5 and using inductionwith respect to k, we have for any integers k ≥ 1 and p > 1,

E[∣∣ξ (1,k)

s,t (u)∣∣pH⊗(k−1)

]< +∞.(A.15)

Then by the same construction and induction arguments, for positive integersk (≥ 1) we can define a sequence of random variables {f (ε)(s, t)}, {ξ (ε,k)

s,t (u)} and{Dkf (ε)(s, t)}. Hence we have completed the proof of Theorem 4.2. �

Acknowledgments. This paper is a revised version of Discussion PaperNo. 98-F-6 at the Faculty of Economics, University of Tokyo. We thank ProfessorsS. Kusuoka and N. Yoshida for some discussions on technical issues involved. Wealso thank a referee and an Associate Editor for their helpful comments on earlierversions. However, we are responsible for any remaining errors in this paper.

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FACULTY OF ECONOMICS

UNIVERSITY OF TOKYO

BUNKYO-KU, HONGO 7-3-1TOKYO 113JAPAN

GRADUATE SCHOOL

OF MATHEMATICAL SCIENCES

UNIVERSITY OF TOKYO

MEGURO-KU, KOMABA 3-8-1TOKYO 153JAPAN

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